I am using a different computer than mine. I cannot get to my Google Account because that's the account of my brother which is registered.
Open https://accounts.google.com/ and Sign in with a different account (a link at the bottom of the page).
Or login to your operating system (Windows?) with a different username. Or just use another browser. Or use incognito mode in Chrome browser.
Depending on the setup you have a couple of options.
If you're only checking this once then you can use "Private Browsing" / "Incognito Mode" in the browser to log in. Once you close the window your session will be ended and the original login will be left in place.
If this is a regular occurrence then you may want to try just using a different browser. E.g. if your brother uses Firefox then you could use Chrome, &c. This way you'll each keep your own logins and settings.
Below your brother's email address, there is an option "add user." Just click it and log in with your own email address.